Penalty-based aggregation of multidimensional data

Research in aggregation theory is nowadays still mostly focused on algorithms to summarize tuples consisting of observations in some real interval or of diverse general ordered structures. Of course, in practice of information processing many other data types between these two extreme cases are worth inspecting. This contribution deals with the aggregation of lists of data points in Rd for arbitrary d≥1. Even though particular functions aiming to summarize multidimensional data have been discussed by researchers in data analysis, computational statistics and geometry, there is clearly a need to provide a comprehensive and unified model in which their properties like equivariances to geometric transformations, internality, and monotonicity may be studied at an appropriate level of generality. The proposed penalty-based approach serves as a common framework for all idempotent information aggregation methods, including componentwise functions, pairwise distance minimizers, and data depth-based medians. It also allows for deriving many new practically useful tools.